The short version is this: The first sentence of the Fourteenth Amendment conveys U.S. citizenship on all persons "born ... in the United States and subject to the jurisdiction thereof." Obviously we are talking here about persons "born ... in the United States." Thus the children of illegal aliens are not U.S. citizens only if they are not "subject to the jurisdiction" of the United States.

But there is no sense in which children of illegal aliens are not "subject to the jurisdiction" of the United States. So long as they remain in the United States, they are subject to U.S. law. If they violate U.S. law, they can be arrested by U.S. law enforcement, brought before a U.S. court, and sentenced to U.S. prison.

Moreover, it's evident from nineteenth-century international law who might be born in the United States but not be "subject to the jurisdiction" of the United States: children of ambassadors and other diplomatic personnel. Under the international law of the time (as is still largely true in modern law) foreign diplomats had immunity from local law. They could not be arrested or brought before a U.S. court. If they violated U.S. law, the U.S.'s sole remedy was to request the sending nation to recall them. They were not, it was said, subject to U.S. jurisdiction. And an ambassador's family enjoyed the same immunity.

The framers of the Fourteenth Amendment also apparently thought members of some Native American tribes were not "subject to the jurisdiction" of the United States for purposes of the Fourteenth Amendment. That was either because the tribes had treaties with the United States that preserved their sovereignty over internal affairs (and thus their members were not subject to U.S law with respect to internal matters) or because the tribes had not yet been defeated militarily and thus were literally beyond the reach of U.S. authority. In either event, the children were like the paradigm children of ambassadors, in the sense that ordinary U.S. legal process did not apply to them. And likewise they were completely unlike modern children of illegal aliens, to whom U.S. legal process applies completely.

It's true, as opponents of birthright citizenship contend, that the framers of the Fourteenth Amendment's first sentence were principally thinking of freed slaves (and probably children of legal immigrants). Illegal immigration was not a major issue in 1868, although it became one fairly soon afterward. What the framers would have thought about birthright citizenship for children of illegal aliens is something about which we can only speculate. But the meaning of the law they wrote is quite clear.